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          Problem 253
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<h1 id="Problem-253"><a href="#Problem-253" class="headerlink" title="Problem 253"></a><a href="https://projecteuler.net/problem=253" target="_blank" rel="noopener">Problem 253</a></h1><hr>
<p><strong>Tidying up</strong></p>
<p>A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers 1 to 40 in order.</p>
<p>Every night, the child’s father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up the pieces at random and places them in the correct order. As the caterpillar is built up in this way, it forms distinct segments that gradually merge together. The number of segments starts at zero (no pieces placed), generally increases up to about eleven or twelve, then tends to drop again before finishing at a single segment (all pieces placed).</p>
<p>For example:</p>
<table>
<thead>
<tr>
<th align="center"><strong>Piece Placed</strong></th>
<th align="center"><strong>Segments So Far</strong></th>
</tr>
</thead>
<tbody><tr>
<td align="center">12</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">29</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">34</td>
<td align="center">5</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">35</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">…</td>
<td align="center">…</td>
</tr>
</tbody></table>
<p>Let M be the maximum number of segments encountered during a random tidy-up of the caterpillar.<br>For a caterpillar of ten pieces, the number of possibilities for each M is</p>
<table>
<thead>
<tr>
<th align="center"><strong>M</strong></th>
<th align="center"><strong>Possibilities</strong></th>
</tr>
</thead>
<tbody><tr>
<td align="center">1</td>
<td align="center">512</td>
</tr>
<tr>
<td align="center">2</td>
<td align="center">250912</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">1815264</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">1418112</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">144000</td>
</tr>
</tbody></table>
<p>so the most likely value of M is 3 and the average value is <sup>385643</sup>⁄<sub>113400</sub> = 3.400732, rounded to six decimal places.</p>
<p>The most likely value of M for a forty-piece caterpillar is 11; but what is the average value of M?</p>
<p>Give your answer rounded to six decimal places.</p>
<hr>
<p><strong>清理</strong></p>
<p>小朋友有一个“数字毛毛虫”玩具，包含有40片拼板，分别标号；如果把它们都拼起来，将会组成一条直线，且按照1到40顺序排列。</p>
<p>每天晚上，小朋友的爸爸都要把玩具房里撒了一地的毛毛虫拼板捡起来。他捡的时候是完全随机的，捡起来之后，再按照正确的顺序拼好。这样一来，毛毛虫拼板将会构成分离的片段，并且不断合并直到组成完整的毛毛虫。片段数从0开始（没有捡起任何一块拼板），不断上升到大约11或12，然后再次下降，直到最终只有一段（所有的拼板都组合起来了）。</p>
<p>例如：</p>
<table>
<thead>
<tr>
<th align="center"><strong>捡起的拼板标号</strong></th>
<th align="center"><strong>目前为止的片段数</strong></th>
</tr>
</thead>
<tbody><tr>
<td align="center">12</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">29</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">34</td>
<td align="center">5</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">35</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">…</td>
<td align="center">…</td>
</tr>
</tbody></table>
<p>记M是这个随机地清理毛毛虫过程中遇到的最大片段数。<br>若毛毛虫分成10片，出现不同M的可能情况数分别是：</p>
<table>
<thead>
<tr>
<th align="center"><strong>M</strong></th>
<th align="center"><strong>可能情况</strong></th>
</tr>
</thead>
<tbody><tr>
<td align="center">1</td>
<td align="center">512</td>
</tr>
<tr>
<td align="center">2</td>
<td align="center">250912</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">1815264</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">1418112</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">144000</td>
</tr>
</tbody></table>
<p>因此，最可能出现的M值为3，M的平均值为<sup>385643</sup>⁄<sub>113400</sub> = 3.400732，此处四舍五入到六位小数。</p>
<p>当毛毛虫共有40片时，最可能出现的M值为11；那么M的平均值是多少？</p>
<p>将你的答案四舍五入到六位小数。</p>
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